Q: A company’s price-demand function is $$p(x) = 75 - 3x$$, where $$1 \leq x \leq 20. $$What is the estimated price per chip when the demand is 11 million chips?

$$p(11) = 75 - 3 \times 11 = 42$$

Q: Using the depreciation equation $$V = 224,000 - 15,500t$$, what is the value of the bulldozer after $$12$$ years?

$$V = 224,000 - 15,500 \times 12 = 38,000$$

Q: Find the value of an investment of $$10,000 in$$ $$12$$ years if it earns a rate of $$3.95\%$$ compounded continuously.

$$A = 10,000 \times e$$^{0.0395 \times 12}$$ \approx 15,800$$

Q: Suppose $$2,500 is $$invested at $$7\%$$ compounded quarterly. How much money will be in the account in $$15$$ years?

$$A = 2,500 \times (1 + \frac{0.07}{4}$$)^{4 \times 15}$$ \approx 7,000$$

Q: A couple wants to have $$40,000$$ for their child’s education in $$17$$ years. How much should they invest now at $$5.5\%$$ compounded daily? (Round to the nearest dollar.)

$$P = \frac{40,000}{(1 + \frac{0.055}{365}$$)^{365 \times 17}$$} \approx 16,800$$

Question 1

Which of the following represents the domain of the function $$F(x) = 2x^3$ - x^2$ + 3$$?

Accepted Answer

All real numbers $$x$$

Question 2

A company’s price-demand function is $$p(x) = 75 - 3x$$, where $$1 \leq x \leq 20. $$What is the estimated price per chip when the demand is 11 million chips?

Accepted Answer

$$p(11) = 75 - 3 	imes 11 = 42$$

Question 3

Given the price-demand function $$p(x) = 75 - 3x$$, what is the company’s revenue function $$R(x)$$?

Accepted Answer

$$R(x) = x \cdot (75 - 3x)$$

Question 4

At a price of $$2.28$$ per bushel, the supply of barley is $$7,500$$ million bushels. At a price of $$2.37$$ per bushel, the supply is $$7,900$$ million bushels. What is the price-supply equation of the form $$p = mx + b$$?

Accepted Answer

Question 5

A bulldozer was purchased for $$224,000 and has a depreciated value of 100$$,000$$ after 8$ years. Assuming linear depreciation, what is the equation relating value V$ to time t$?

Accepted Answer

$$V = 224,000 - 15,500t$$

Question 6

Using the depreciation equation $$V = 224,000 - 15,500t$$, what is the value of the bulldozer after $$12$$ years?

Accepted Answer

$$V = 224,000 - 15,500 	imes 12 = 38,000$$

Question 7

A company manufacturing surfboards has fixed costs of $$300$$ per day and total costs of $$5,100$$ per day at a daily output of $$20$$ boards. Assuming linearity, what is the cost function $$C(x)$$?

Accepted Answer

$$C(x) = 300 + 240x$$

Question 8

Given the cost function $$C(x) = 300 + 240x$$, what is the average cost function $$C_a(x)$$?

Accepted Answer

$$C_a(x) = \frac{300 + 240x}{x}$$

Question 9

Find the value of an investment of $$10,000 in$$ $$12$$ years if it earns a rate of $$3.95\%$$ compounded continuously.

Accepted Answer

$$A = 10,000 	imes e$$^{0.0395 	imes 12}$$ \approx 15,800$$

Question 10

Suppose $$2,500 is $$invested at $$7\%$$ compounded quarterly. How much money will be in the account in $$15$$ years?

Accepted Answer

$$A = 2,500 	imes (1 + \frac{0.07}{4}$$)^{4 	imes 15}$$ \approx 7,000$$

Question 11

A couple wants to have $$40,000$$ for their child’s education in $$17$$ years. How much should they invest now at $$5.5\%$$ compounded daily? (Round to the nearest dollar.)

Accepted Answer

$$P = \frac{40,000}{(1 + \frac{0.055}{365}$$)^{365 	imes 17}$$} \approx 16,800$$

Question 12

Assume money invested in the Gabelli Growth Fund earns $$21.36\%$$ compounded annually. How long will it take for the investment to double?

Accepted Answer

$$t = \frac{\ln 2}{\ln(1 + 0.2136)} \approx 4$$ years

Question 13

If the price-demand function is $$p(x) = 75 - 3x$$ and the revenue function is $$R(x) = x(75 - 3x)$$, what is the maximum revenue the company can earn?

Accepted Answer

Maximum revenue occurs at $$x = \frac{75}{6}$$, so $$R(12.5) = 12.5 	imes (75 - 3 	imes 12.5) = 12.5 	imes 37.5 = 468.75$$

Question 14

Given the price-supply equation $$p = 0.000225x + 0.585$$ and the price-demand equation $$p = -0.0001x + 3.08$$, what is the equilibrium point?

Accepted Answer

Set $$0.000225x + 0.585 = -0.0001x + 3.08$$, solve for $$x$$: $$x = \frac{3.08 - 0.585}{0.000225 + 0.0001} = 11,080$$ million bushels, $$p = 3.08 - 0.0001 	imes 11,080 = 1.972$$

Question 15

If the average cost function is $$C_a(x) = \frac{300 + 240x}{x}$$, what happens to the average cost per board as $$x$$ increases?

Accepted Answer

The average cost approaches $$240 as$$ $$x$$ increases.

Question 16

Which of the following best describes the effect of compounding frequency on investment growth?

Accepted Answer

The more frequently interest is compounded, the greater the final amount, due to more periods of interest accumulation.

Question 17

A company’s cost function is $$C(x) = 300 + 240x. If $$the company produces $$25$$ boards in a day, what is the average cost per board?

Accepted Answer

$$C_a(25) = \frac{300 + 240 	imes 25}{25} = \frac{6,300}{25} = 252$$

Question 18

If $$f(x) = x - 2x + 4$$, what is the simplified form of $$f(x)$$?

Accepted Answer

$$f(x) = -x + 4$$

Chapter 1 Business Calculus: Functions, Linear Models, and Financial Applications – Guided Study

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